Let F(x) be an indefinite integral of ^2 x. Assertion (A) : The f… - Vidyadip

MCQ

Let $F(x)$ be an indefinite integral of $\sin ^2 x$.
Assertion (A) : The function $F(x)$ satisfies $F(x+\pi)=F(x)$ for all real $x$.
Reason (R) : $\sin ^2(x+\pi)=\sin ^2 x$ for all real $x$.

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

✓

Answer

$
\begin{aligned}
(d): F(x) & =\int \sin ^2 x d x=\int \frac{1}{2}(1-\cos 2 x) d x \\
& =\frac{x}{2}-\frac{\sin 2 x}{4}+C
\end{aligned}
$
$
\because \quad F(x+\pi)-F(x)=\frac{\pi}{2} \neq 0
$
$\therefore \quad$ Assertion is false.
$
\sin ^2(x+\pi)=(-\sin x)^2=\sin ^2 x
$
$\therefore \quad$ Reason is true.

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